Newton's method The following sequences come from the...

Newton's method The following sequences come from the recursion formula for Newton's method, $$x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}.$$ Do the sequences converge? If so, to what value? In each case, begin by identifying the function $f$ that generates the sequence. a. $x_{0}=1, \quad x_{n+1}=x_{n}-\frac{x_{n}^{2}-2}{2 x_{n}}=\frac{x_{n}}{2}+\frac{1}{x_{n}}$ b. $x_{0}=1, \quad x_{n+1}=x_{n}-\frac{\tan x_{n}-1}{\sec ^{2} x_{n}}$ c. $x_{0}=1, \quad x_{n+1}=x_{n}-1$

Newton's method The following sequences come from the recursion formula for Newton's method,
$$x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}.$$
Do the sequences converge? If so, to what value? In each case, begin by identifying the function $f$ that generates the sequence.
a. $x_{0}=1, \quad x_{n+1}=x_{n}-\frac{x_{n}^{2}-2}{2 x_{n}}=\frac{x_{n}}{2}+\frac{1}{x_{n}}$
b. $x_{0}=1, \quad x_{n+1}=x_{n}-\frac{\tan x_{n}-1}{\sec ^{2} x_{n}}$
c. $x_{0}=1, \quad x_{n+1}=x_{n}-1$

Key Concepts

Newton's Method

Newton's method is an iterative procedure used to approximate the roots of a nonlinear equation. It uses the derivative of a function along with the function itself to quickly update an initial guess, yielding a sequence of approximations that converge to a zero of the function, assuming certain conditions on the function and its derivatives are met.

Recursion Formula

The recursion formula, x??? = x? - f(x?)/f'(x?), provides a systematic way to generate successive approximations to the root of the function. Each iteration uses the current estimate, the value of the function at that estimate, and the value of its derivative, thereby 'correcting' the estimate based on the local linear behavior of the function.

Convergence of Iterative Sequences

Convergence in the context of iterative methods refers to the tendency of a sequence of approximations to approach a specific value, which is the desired root of the equation. The rate and reliability of convergence depend on factors such as the nature of the function, the quality of the initial guess, and the behavior of the derivative near the root.

Identification of the Function f

In applying Newton's method, it is crucial to correctly identify the function f whose root is sought. Doing so determines the terms in the iteration formula. Once f is defined, its derivative must be computed in order for the method to be implemented, making the identification of f a fundamental step in the process.

Fixed Points

A fixed point is a value that remains unchanged under the iteration. In Newton's method, a fixed point occurs when the updated approximation equals the current one, which implies that f(x) is zero. Thus, the fixed point represents the root of the equation, and showing that the iteration has a fixed point can confirm the success of the method.

Stability and Divergence in Iterative Methods

The analysis of stability involves determining whether small deviations from a fixed point diminish or grow under the iteration. Newton's method converges rapidly when the function is well-behaved near the root, but it may diverge or oscillate if the derivative is zero, nearly zero, or the initial guess is too far from the true root. Understanding these stability criteria is key to assessing the method's effectiveness in practical applications.

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